Question

\( \sin 765^{\circ} \) is equal to:

• A. 1
• B. 0
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( \frac{1}{2} \)
• E. \( \frac{1}{\sqrt{2}} \)

Hint:

For very large angles, keep subtracting $360$ until you get a value between $0$ and $360$. If the value is in the 2nd, 3rd, or 4th quadrant, use ASTC rules to find the reference angle.

Answer 1

Answer: (E)

Concept: Trigonometric functions are periodic. The sine function has a period of \( 360^{\circ} \), meaning \( \sin(\theta + n \cdot 360^{\circ}) = \sin \theta \) for any integer \( n \). To solve this, we find the remainder when the angle is divided by \( 360^{\circ} \).

Step 1: Reduce the angle by removing full rotations.
We divide \( 765^{\circ} \) by \( 360^{\circ} \): \[ 765^{\circ} = 2 \times 360^{\circ} + 45^{\circ} \] Since \( 2 \times 360^{\circ} = 720^{\circ} \), the angle \( 765^{\circ} \) is coterminal with \( 45^{\circ} \).

Step 2: Evaluate the sine of the reduced angle.
Using the periodicity property: \[ \sin 765^{\circ} = \sin(2 \times 360^{\circ} + 45^{\circ}) = \sin 45^{\circ} \] From the standard trigonometric table: \[ \sin 45^{\circ} = \frac{1}{\sqrt{2}} \]